A spanning tree is a subgraph of a connected graph that connects all vertices together using the fewest possible number of edges without cycles. There are two common algorithms for finding a minimum spanning tree: Kruskal's algorithm and Prim's algorithm. Both algorithms find a subset of edges that forms a tree connecting all vertices with minimum total edge weight. Kruskal's algorithm considers all edges from lowest to highest weight, while Prim's algorithm starts from a single vertex and incrementally adds adjacent vertices.

1. 1
Spanning Trees

2. 2
Graph - Spanning Tree
• A tree that contains every vertex of a connected graph is
called spanning tree. It is an undirected tree consisting
of only those edges that are necessary to connect all the
vertices of the original graph G.
• CHARACTERISTICS
1. It is a subgraph of a graph G that contain all the
vertex of graph G.
2. For any pair of vertices, there exists only one path
between them.
3. It is a connected graph with no cycles.
4. A graph may have many spanning tree.

3. 3
EXAMPLE

4. 4
Graph - Minimum Spanning Tree
• Let G=(V,E,W) be any weighted graph. Then a
spanning tree whose cost is minimum is
called minimum spanning tree. The cost of the
spanning tree is defined as the sum of the
costs of the edges in that tree.

5. 5
Kruskal's algorithm
• It is an algorithm in graph theory that finds a minimum
spanning tree for a connected weighted graph.
• It finds a subset of the edges that forms a tree that
includes every vertex, where the total weight of all the
edges in the tree is minimized.
• If the graph is not connected, then it finds a minimum
spanning forest (a minimum spanning tree for each
connected component).
• Kruskal's algorithm is an example of a greedy algorithm.

6. 6
1. Create a forest F (a set of trees), where each vertex in
the graph is a separate tree
2. Create a set S containing all the edges in the graph
3. Repeat Step 4 and 5, while S is nonempty and F is not
yet spanning
4. Remove an edge with minimum weight from S
5. IF that edge connects two different trees, then add it to
the forest, combining two trees into a single tree ELSE
discard that edge.
6. Exit
At the termination of the algorithm, the forest has only
one component and forms a minimum spanning tree of
the graph.

7. 7
Example of Kruskal’s Algorithm
This is our original graph. The numbers near the
arcs indicate their weight. None of the arcs are
highlighted.

8. 8
AD and CE are the shortest
arcs, with length 5, and AD
has been arbitrarily chosen,
so it is highlighted.
CE is now the shortest arc
that does not form a cycle,
with length 5, so it is
highlighted as the second
arc.

9. 9
The next arc, DF with length 6,
is highlighted using much the
same method.
The next-shortest arcs are AB and BE,
both with length 7. AB is chosen
arbitrarily, and is highlighted. The arc
BD has been highlighted in red,
because there already exists a path
(in green) between B andD, so it
would form a cycle (ABD) if it were
chosen.

10. 10
The process continues to
highlight the next-smallest arc,
BE with length 7. Many more arcs
are highlighted in red at this
stage: BC because it would form
the loop BCE, DE because it
would form the loop DEBA, and
FE because it would form FEBAD.
Finally, the process finishes with
the arc EG of length 9, and the
minimum spanning tree is found.

11. 11
Prim's algorithm
• It is a greedy algorithm that finds a minimum
spanning tree for a connected weighted
undirected graph.
• Same as in Kruskal’s Algorithm, it finds a
subset of the edges that forms a tree that
includes every vertex, where the total weight
of all the edges in the tree is minimized.
• It is also sometimes called the DJP algorithm,
the Jarnik algorithm, or the Prim–Jarnik
algorithm.

12. 12
Prim's algorithm (contd..)
• The only spanning tree of the empty graph
(with an empty vertex set) is again the empty
graph.
• The following description assumes that this
special case is handled separately.
• The algorithm continuously increases the size
of a tree, one edge at a time, starting with a
tree consisting of a single vertex, until it spans
all vertices.

13. 13
• Input: A non-empty connected weighted graph with
vertices V and edges E (the weights can be negative).
Algorithm:
1. Initialize: Vnew = {x}, where x is an arbitrary node (starting
point) from V, Enew = { }
2. Repeat Step 3 and 4, until Vnew = V
3. Choose an edge (u, v) with minimal weight such that u is
in Vnew and v is not (if there are multiple edges with the
same weight, any of them may be picked)
4. Add v to Vnew, and (u, v) to Enew
5. Exit
• Output: Vnew and Enew describe a minimal spanning tree

14. 14
Example of Prim’s Algorithm
This is our original weighted graph. The numbers
near the edges indicate their weight.

15. 15
Vertex D has been arbitrarily
chosen as a starting point. Vertices
A, B, E and F are connected to D
through a single edge. A is the
vertex nearest to D and will be
chosen as the second vertex along
with the edge AD.
The next vertex chosen is the
vertex nearest to either D or A. B is
9 away from D and 7 away
from A, E is 15, and F is 6. F is the
smallest distance away, so we
highlight the vertex F and the
arc DF.

16. 16
The algorithm carries on as
above. Vertex B, which is 7
away from A, is highlighted.
In this case, we can choose
between C, E, and G. C is 8
away from B, E is 7 away
from B, and G is 11 away
from F. E is nearest, so we
highlight the vertex E and
the arc BE.

17. 17
Here, the only vertices
available are C and G. C is 5
away from E, and G is 9 away
from E. C is chosen, so it is
highlighted along with the
arc EC.
Vertex G is the only
remaining vertex. It is 11
away from F, and 9 away
from E. E is nearer, so we
highlight it and the arc EG.

18. 18
Now all the vertices have
been selected and the
minimum spanning tree is
shown in green. In this case,
it has weight 39.

19. 19
U Edge(u,v) V - U
{ } {A,B,C,D,E,F,G}
{D}
(D,A) = 5 V
(D,B) = 9
(D,E) = 15
(D,F) = 6
{A,B,C,E,F,G}
{A,D}
(D,B) = 9
(D,E) = 15
(D,F) =6 V
(A,B) = 7
{B,C,E,F,G}
{A,D,F}
(D,B) = 9
(D,E) = 15
(A,B) = 7 V
(F,E) = 8
(F,G) = 11
{B,C,E,G}
{A,B,D,F}
(B,C) = 8
(B,E) = 7 V
(D,B) = 9 cycle
(D,E) = 15
(F,E) = 8
(F,G) = 11
{C,E,G}
{A,B,D,E,F}
(B,C) = 8
(D,B) = 9 cycle
(D,E) = 15 cycle
(E,C) = 5 V
(E,G) = 9
(F,E) = 8 cycle
(F,G) = 11
{C,G}
{A,B,C,D,E,F}
(B,C) = 8 cycle
(D,B) = 9 cycle
(D,E) = 15 cycle
(E,G) = 9 V
(F,E) = 8 cycle
(F,G) = 11
{G}
{A,B,C,D,E,F,G}
(B,C) = 8 cycle
(D,B) = 9 cycle
(D,E) = 15 cycle
(F,E) = 8 cycle
(F,G) = 11 cycle
{ }